Theorem ntrclscls00 42493

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclscls00	⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑗,𝐾,𝑘   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclscls00

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	1, 2, 3	ntrclsfv1 42482	. . . . 5 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
5	4	fveq1d 6864	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐾‘∅))
6	2, 3	ntrclsbex 42461	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	1, 2, 3	ntrclsiex 42480	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8		eqid 2731	. . . . 5 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
9		0elpw 5331	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝐵
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝒫 𝐵)
11		eqid 2731	. . . . 5 ⊢ ((𝐷‘𝐼)‘∅) = ((𝐷‘𝐼)‘∅)
12	1, 2, 6, 7, 8, 10, 11	dssmapfv3d 42446	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
13	5, 12	eqtr3d 2773	. . 3 ⊢ (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14		dif0 4352	. . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵
15	14	fveq2i 6865	. . . . . 6 ⊢ (𝐼‘(𝐵 ∖ ∅)) = (𝐼‘𝐵)
16		id 22	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘𝐵) = 𝐵)
17	15, 16	eqtrid 2783	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
18	17	difeq2d 4102	. . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵 ∖ 𝐵))
19		difid 4350	. . . 4 ⊢ (𝐵 ∖ 𝐵) = ∅
20	18, 19	eqtrdi 2787	. . 3 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
21	13, 20	sylan9eq 2791	. 2 ⊢ ((𝜑 ∧ (𝐼‘𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22		pwidg 4600	. . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
23	6, 22	syl 17	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
24	1, 2, 3, 23	ntrclsfv 42486	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))))
25	19	fveq2i 6865	. . . . . 6 ⊢ (𝐾‘(𝐵 ∖ 𝐵)) = (𝐾‘∅)
26		id 22	. . . . . 6 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
27	25, 26	eqtrid 2783	. . . . 5 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘(𝐵 ∖ 𝐵)) = ∅)
28	27	difeq2d 4102	. . . 4 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = (𝐵 ∖ ∅))
29	28, 14	eqtrdi 2787	. . 3 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = 𝐵)
30	24, 29	sylan9eq 2791	. 2 ⊢ ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼‘𝐵) = 𝐵)
31	21, 30	impbida 799	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3459   ∖ cdif 3925  ∅c0 4302  𝒫 cpw 4580   class class class wbr 5125   ↦ cmpt 5208  ‘cfv 6516  (class class class)co 7377   ↑_m cmap 8787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-1st 7941  df-2nd 7942  df-map 8789

This theorem is referenced by: (None)